I will defer to others answering the parts of your question concerning the relationship between Markov processes and martingales (@SRKX has already given a good explanation of the relationship) and concerning statistical testing. Broadly, however, it is not possible to "prove" either assumption, but only to fail to reject them. A Non-Random Walk Down Wall Street by Andrew Lo and Craig MacKinlay provides many examples of the sophisticated statistical techniques which have been used to disprove these assumptions.

I focus here on the realism of the two assumptions.

To say that stock prices are a martingale is to essentially say they are weak-form efficient (see Efficient-Market Hypothesis, or EMH). Weak-form efficiency says that knowledge of all past prices is not informative regarding the expectation of future prices. A martingale is a special case of weak-form efficiency which says that the expected next future price is equal to current price. This is either nearly true when examining a sufficiently short horizon or it is precisely true when considering the "discounted price process," which discounts the price by the risk-free rate plus the equity risk premium.

The EMH was taken very seriously and much evidence was accumulated in its favor in the 1960s and 1970s. Starting in the 1980s and particularly in the 1990s, evidence began accumulating of ways in which stock prices violate the EMH (see Behavioral Finance), particularly the discovery of short-horizon reversal, medium-horizon momentum, and long-horizon overreaction. Nevertheless, these effects are relatively weak, and the assumption of weak-form efficiency is generally seen as reasonable outside of a few specific instances.

To say that stock prices have the Markov property is to assume much greater stability in the data-generating process than is generally believed to be the case. In particular, if prices were a Markov process, then knowledge of merely the current price would be a sufficient statistic for the probability distribution of future prices. A clear example of how this is not true is the demonstrated persistence of volatility. In other words, knowing the volatility of past prices adds significant information regarding the future probability distribution of prices relative to just knowing the current price.

In short, stock prices are neither martingales nor Markovian, but the former is a much better working assumption than the latter.

$\begingroup$@Andrei I think you confused my explanation of EMH with my explanation of martingales, which I have now clarified. Sorry for the imprecise English.$\endgroup$
– Tal FishmanSep 12 '11 at 14:45

Martingale and Markov process are both stochastic processes where the sequences of random variables are not entirely independent, and their differences are:

In martingale, the expectation of the next value IS the present value, so this property is sometimes called 'fair game'.

In Markov process, the expectation of the next value only DEPENDS ON the present value. In others words, the future of the process is solely based upon the present state, not on the sequence of events that preceded it, so the Markov property is 'memoryless'.

I agree with the others Markov does not imply martingale and vice versa.

There are many sources on empirical tests for these properties.

In my opinion these assumptions are not unreasonable. Markov property only says that all past information about the stock price process (historical prices, historical volume, etc.) is incorporated in the current price and therefore only the current price is relevant. I believe that it is logical to assume that publicly available historical information is already priced in. For instance even the anomalies violating weak form efficiency (e.g. the January effect) tend to disappear over time as market participants trade on the information thereby incorporating the information into the price. Assuming that the stock price process is a submartingale only says that in expectation the future stock price should be greater than or equal to today's price. Intuitively, investors would not participate (long positions) in the stock market if prices were expected to decline. Take stock price process
$$
dX = \alpha Xdt + \sigma XdW
$$
A submartingale implies $\alpha \geq 0$

For most assets I don't believe that that is an unreasonable assumption.

A martingale process simply implying the expections of future is the same as present state, if the stock price process are martingales then a collection of stocks over a long time should not show the trends we saw now in the stock market.

A markov process is a process where future is independent of the past, again, not likely, at the very least, stock price movement is a result of supply and demand with performance expection adjustments, if it is a markov process then the stock holder should make the same kind of decisions despite of how much the stock he and the investment combinations he hold and, yet we always try to make different kind of decsions under different conditions that try to optimizing our returns, well, if you hold most of the tradeable stock of a listed company, you can even largely determine the price movement of the stock and this is something many investment agencies have been doing all the time.

The two are all too elemental to represent stock price process, althrough the two were often used (althrough with somewhat doubtful success since there are better alternatives out there) as compoents to build stock and other more complicate process/time series models.